00-69 Michael Christ, Alexander Kiselev

WKB Asymptotics of Generalized Eigenfunctions of One-Dimensional Schr\"odinger Operators (51K, LaTeX) Feb 10, 00

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Abstract. We prove the WKB asymptotic behavior of solutions of the differential equation $-d^2u/dx^2+V(x)u=k^2u$ in two cases. First, for a.e.\ $k^2$ when $V \in L^p(\reals)$, where $1 \leq p <2$. Second, for a.e.\ $k^2>A$ when $V \in L^{\infty}(\reals)$ and $V' \in L^p(\reals)$, $1 \leq p <2$, where $A = \limsup_{x \rightarrow \infty}V(x)$. These results imply that Schr\"odinger operators with such potentials have absolutely continuous spectrum on $(0, \infty)$ ($(A, \infty)$ in the second case). We also establish WKB asymptotic behavior of solutions for some energy-dependent potentials.

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